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The Evolution of Mathematical Thinking in Chinese Mathematics Education

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mathematics Article The Evolution of Mathematical Thinking in Chinese Mathematics Education Na Li 1,*, Ida Ah Chee Mok 2 and Yiming Cao 3,4,* 1 School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China 2 Faculty of Education, The University of Hong Kong, Hong Kong, China; [email protected] 3 School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China 4 International Center for Research in Mathematics Education, Beijing Normal University, Beijing 100875, China * Correspondence: [email protected] (N.L.); [email protected] (Y.C.); Tel.: +86-10-6786-2762 (N.L.) Received: 27 February 2019; Accepted: 20 March 2019; Published: 24 March 2019 Abstract: Mathematical thinking (MT) has been one of the most important goals for mathematics education as it can support sustainable mathematics learning. Its role in school mathematics has recently been explicitly identified as one of “Four Basics” in the latest national curriculum standard for compulsory education, which is seen as one of the prominent features of Chinese mathematics education. This paper reviewed definitions, descriptions, and explanations from a historical perspective and aimed to provide a comprehensive and contemporary conceptualization for MT in a Chinese context, which can be applied as a comparison to MT in English works. To achieve this, document analysis was applied in this study. Major official documents, papers, and books were reviewed to see the process of MT conceptualization given by the policy makers and researchers. Results indicated that MT places more emphasis on the process of mathematical methods application in problem solving, such as the method of combination of symbolic and graphic mathematics. Mathematical thought is also recommended by Chinese researchers to help students think like mathematicians. Another major characteristic is that the classification of major types of MT is usually focused on that which can make the concept more understandable. Keywords: mathematical thinking; mathematical method; Four Basics; curriculum standard; China 1. Introduction Mathematical thinking (MT) is essential for teaching and learning mathematics [1]. It is not only the foundation for children’ development [2] but also the cornerstone of students’ sustainable development in mathematics [3]. Studies have shown that Asian mathematics teachers used to explain mathematics subjects matter in more depth and place a greater stress on thinking and understanding [4, 5]. Especially in China (in this study, mainland China is considered), mathematical thinking (MT) is paid much attention in classroom instruction, which can be seen as the most important feature of Chinese mathematics education [6]. Hence, the stress on MT can help to explain Chinese students’ good outstanding performance in international assessment [7–10], such as the Trends in International Mathematics and Science Study (TIMSS) and the Programme for International Student Assessment (PISA). While mathematics education in East Asian countries, such as Singapore and Japan, has been devoted significant attention, however, much less attention has been paid to Chinese mathematics education [11]. Consequently, introducing the unique characteristics in Chinese mathematics education would open a door for others to know more about Chinese mathematics education. The primary goal of this study is to provide a comprehensive understanding of mathematical thinking based on the statements of MT in the official documents. Mathematics 2019, 7, 297; doi:10.3390/math7030297 www.mdpi.com/journal/mathematics Mathematics 2019, 7, 297 2 of 18 In mainland China, a center-periphery curriculum development system is followed, and nationwide unified and mandatory curriculum standards (or syllabus) are used as guidelines for all teaching and learning activities at different grade levels and serve as a direct channel for major education reforms [12,13]. In the latest version of the Compulsory Education Mathematics Curriculum Standard issued in 2011 [14], one of the most important changes was the extension of “Two Basics” (here refers to basic knowledge and skills) as “Four Basics” [15]. This can be seen as a great achievement of basic education curriculum reform after entering the 21st Century in China [16]. Moreover, while “Two Basics” was considered to be a unique feature in Chinese mathematics education in the past [17,18], “Four Basics” is now seen as the unique feature of mathematics curriculum reform in China [15]. The principle of “Two Basics” was generalized to describe the characteristics of Chinese mathematics education. It literally refers to basic knowledge and basic skills [19,20]. Mathematical concepts, rules, formulas, axioms, and theorems can be included in basic knowledge, whereas, basic skills include computation, data processing and the use of calculators, simple reasoning, and drawing tables and figures [19]. Moreover, basic knowledge is mainly attained through memorization, while basic skills are primarily acquired by practice [21]. For example, manipulation on polynomial expressions, algebraic fractions, exponential and radical expressions, and memorization of the rules [22]. The final purpose was to help students solve problems, especially the examination questions, accurately and speedily [22]. For instance, when teaching the “distributive law of multiplication over addition”, students are required to remember the formula “a (b + c) = ab + ac” accurately and given a number of exercises to practice and memorize the law and calculate fast. Mathematical thinking can be made more condensed and faster and reach a higher level through skillful calculations and memorization of formulas [22]. In the new curriculum standard [14], basic mathematical thinking and basic experience of mathematics activities were added to make up for “Four Basics”. From the above explanations, it can be seen that the role of MT has been in a prominent position in the latest Curriculum Standard, and not in the previous ones. Why is it that one of the “Four Basics” is in the new Curriculum Standard? To understand it, this study aimed to address the following questions: (a) how is MT developed in these official documents? (b) How are the interpretations for MT shown in these documents? (c) How are the interpretations given by researchers based on these documents? 2. Research Method The document analysis method is “ ... a systematic procedure for reviewing or evaluating documents—both printed and electronic (computer-based and Internet-transmitted material)” [23]. Researchers can apply it to their studies in order to reveal the meaning of the documents, develop understanding and discover new insights related to their research problems [24]. Specifically, researchers can understand the historical roots of specific issues and gain new visions to current phenomena by investigating the documents that bore witness to past events [23]. Documents can not only show the historical roots of specific issues, but they can also highlight specific changes and developments which can be traced by comparing various documents [23]. Documents analyses are usually broad in scope and can cover lengthy time spans while addressing, numerous events and settings [13]. Hence, the document analysis method is appropriate for the first research question, which seeks to understand the evolution of mathematical thinking from a historical perspective. 2.1. Data Selection Document analyses require data selection rather than data collection [13]. This section describes how the documents were selected for analysis. Two aspects were considered: the Chinese key words necessary to identify materials on mathematical thinking and data resource considerations. Mathematics 2019, 7, 297 3 of 18 2.1.1. Keywords in Chinese Determined Before looking into Chinese literature on MT, it was necessary to identify the Chinese key words relevant to it. Once these were established, this study could then discuss the meaning and evolution of mathematical thinking within a Chinese context. Besides Sixiang being used to describe MT, there are still other relevant terms used frequently, such as Shuxuesixiangfangfa (the mathematical method of mathematical thinking/way), and Shuxuesixiang & Fangfa (mathematical thinking and method). These terms have always been used interchangeably in the literature. Considering this, these termswere selected as the key words for the purposes of document selection. The following section describes the data sources and the finalized data selection. 2.1.2. Data Sources The literature was searched primarily through the following two ways. The first was a search of the China National Knowledge Infrastructure (CNKI) database, where almost all educational academic papers could be found. The other was a search through related books on MT. Since the 1980s, a number of important changes have taken place in the construction of Chinese databases. After nearly 30‘years, three main large databases: China National Knowledge Infrastructure (CNKI), WANGFANG DATA, and CQVIP (formally known as Database Research Center under Chongqing Branch of Institute of Scientific & Technical Information of China (CB-ISTIC)) have been established and are both very popular and easy to access. Table1 (below) provides a quick comparison of the three databases. Table 1. Comparison of the three major Chinese databases. Source Database CNKI WANGFANG CQVIP China Academic Journals Full-text China Digital Journals China science and Databases (1979); China Academic Academic
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